polynomial analogon for Fermat’s last theorem

For polynomials with complex coefficients, there is an analogon of Fermat’s last theorem.  It can be proven quite elementarily by using Mason’s theorem (1983), but the original proof (about in 1900) was based on methods of algebraic geometry.

Theorem.  For an integer $n$ greater than 2, there exist no non-constant coprime polynomials $x(t)$, $y(t)$, $z(t)$ in the ring $ℂ[t]$ satisfying

${[x(t)]}^n+{[y(t)]}^n={[z(t)]}^n.$ (1)

Remark.  For  $n=2$, the equation (1) is in e.g. as

$${(2t)}^2+{(1-t^2)}^2={(1+t^2)}^2.$$

Title	polynomial analogon for Fermat’s last theorem
Canonical name	PolynomialAnalogonForFermatsLastTheorem
Date of creation	2013-03-22 19:13:05
Last modified on	2013-03-22 19:13:05
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Theorem
Classification	msc 12E05
Classification	msc 11C08
Synonym	Fermat’s last theorem for polynomials
Related topic	MasonsTheorem
Related topic	WeierstrassSubstitutionFormulas